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A361280 Expansion of e.g.f. exp(x * (1+x)^4).

%I A361280 #25 Nov 11 2023 09:59:54
%S A361280 1,1,9,61,481,4881,55321,682669,9343041,139078081,2216425321,
%T A361280 37736834301,683184324769,13064452686481,262867726142841,
%U A361280 5549111222344621,122499654278797441,2819926900630750209,67539541277010100681,1679557316488693881661
%N A361280 Expansion of e.g.f. exp(x * (1+x)^4).
%H A361280 Winston de Greef, <a href="/A361280/b361280.txt">Table of n, a(n) for n = 0..476</a>
%F A361280 a(n) = n! * Sum_{k=0..n} binomial(4*k,n-k)/k!.
%F A361280 a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * binomial(4,k-1) * a(n-k)/(n-k)!.
%F A361280 D-finite with recurrence a(n) -a(n-1) +8*(-n+1)*a(n-2) -18*(n-1)*(n-2)*a(n-3) -16*(n-1)*(n-2)*(n-3)*a(n-4) -5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Mar 12 2023
%F A361280 a(n) ~ 5^(n/5 - 1/2) * n^(4*n/5) * exp(-256/15625 - 249*5^(4/5)*n^(1/5)/78125 + 236*5^(3/5)*n^(2/5)/9375 + 22*5^(2/5)*n^(3/5)/125 + 4*5^(-4/5)*n^(4/5) - 4*n/5) * (1 + 15409886*5^(1/5)/(48828125*n^(1/5))). - _Vaclav Kotesovec_, Nov 11 2023
%p A361280 A361280 := proc(n)
%p A361280     n!*add(binomial(4*k,n-k)/k!,k=0..n) ;
%p A361280 end proc:
%p A361280 seq(A361280(n),n=0..60) ; # _R. J. Mathar_, Mar 12 2023
%o A361280 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x*(1+x)^4)))
%o A361280 (PARI) a(n) = n!*sum(k=0, n, binomial(4*k, n-k)/k!);
%o A361280 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*binomial(4, j-1)*v[i-j+1]/(i-j)!)); v;
%Y A361280 Column k=4 of A361277.
%Y A361280 Cf. A361283.
%K A361280 nonn
%O A361280 0,3
%A A361280 _Seiichi Manyama_, Mar 06 2023